Complexities of Erez self-dual normal bases

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Complexities of Erez self-dual normal bases

Stéphane Blondeau da Silva

To cite this version:

Stéphane Blondeau da Silva. Complexities of Erez self-dual normal bases. 2017. �hal-01570340�

Complexities of Erez self-dual normal bases

Blondeau Da Silva St´ ephane July 29, 2017

Abstract

The complexities of self-dual normal bases, which are candidates for the lowest complexity basis of some defined extensions, are determined with the help of the number of all but the simple points in well chosen minimal Besicovitch arrangements. In this article, these values are first compared with the expected value of the number of all but the simple points in a minimal randomly selected Besicovitch arrangement in F

for the first 370 prime numbers d . Then, particular minimal Besicovitch arrangements which share several geometrical properties with the arrange- ments considered to determine the complexity will be considered in two distinct cases.

Introduction

Let q be a prime power, F _q be the field of q elements and n be a positive integer. We consider the Galois group of the extension F _q

/ F _q , which is a cyclic group generated by the Frobenius automorphism Φ : x 7→ x ^q . There exists an α that generates a ”normal” basis for F _q

/ F _q , i.e. a basis consisting of the orbit (α, α ^q , ..., α ^q

) of α under the action of the Frobenius. The difficulty of multiplying two elements of the extension expressed in this basis is measured by the complexity of α, namely the number of non-zero entries in the multiplication- by-α matrix: T r(αα ^q

α ^q

)

0≤i,j≤n−1

, where T r is the trace map from F _q

to F _q ([6, 4.1]). As a large number of zero in this matrix enables faster calculations, finding normal bases with low complexity is a significant issue.

Self-dual normal bases are particular normal bases which verify T r(α ^q

α ^q

) = δ i,j (for 0 ≤ i, j ≤ n − 1), where δ is the Kronecker delta. Arnault et al. in [1]

have identified the lowest complexity of self-dual normal bases for extensions of low degree and have showed that the best complexity of normal bases is often achieved from a self-dual normal basis. In [7], Pickett and Vinatier considered cyclotomic extensions of the rationals generated by d

-th roots of unity, where d is a prime. The construction they use yields a candidate for the lowest com- plexity basis for F _p

/ F _p , where p 6= d is a prime which does not split in the chosen extension. They prove that the multiplication table of this basis can be geometrically interpreted by means of an appropriate minimal Besicovitch arrangement. The complexity of the basis, denoted by C d , is here equal to the number of all but the simple points generated by this arrangement in F _d

.

After a brief overview of the properties this arrangement have, we will com-

pare the complexity C d with the expected value of the number of all but the

simple points in a minimal randomly selected Besicovitch arrangement in F _d

for the first 370 prime numbers d. The expectations will be determined using Blondeau Da Silva’s results in [2]. In a third part, we will consider particular minimal Besicovitch arrangements which share several geometrical properties with the arrangements considered to determine the complexity. We will again compare in this part, for the first 370 prime numbers d, C d with the expected value of the number of all but the simple points in the randomly selected men- tioned above arrangement.

1 The minimal Besicovitch arrangement provid- ing the complexity

Let d be a prime number and F _d be the d elements finite field.

A line, in F _d

, is a one-dimensional affine subspace. A Besicovitch arrange- ment B is a set of lines that contains at least one line in each direction. A minimal Besicovitch arrangement is a Besicovitch arrangement that is the union of exactly d + 1 lines in F _d

(see [2]).

The minimal Besicovitch arrangement considered, brought out by Pickett and Vinatier ([7]), and denoted by L , is composed of d + 1 lines with the following equations:

L a : ax − (a + 1)y − p(a) = 0 for a ∈ F _d , L

: x − y = 0 ,

where p is the following polynomial:

∀x ∈ F _d , p(x) = (x + 1) ^d − x ^d − 1

d . (1)

For d ≥ 5, Pickett and Vinatier ([7]) have proved that under the action of Γ = hι, θi (a group generated by two elements of GL

( F _d ), where ι(x, y) = (y, x) and θ(x, y) = (y − x, −x) for (x, y) ∈ F _d

), this arrangement L always has two orbits of cardinal 3: {L

, L

, L

} and {L

, L

2

, L

}. They have also stated that:

• if d ≡ 1 mod 3, there are one orbit of cardinal 2, {L ω , L ω

}, where ω is a primitive cubic root of unity in F _d and ^d−

orbits of cardinal 6;

• if d ≡ 2 mod 3, there are ^d−

orbits of cardinal 6.

The Comp lib 1.1 package have been implemented in Python 3.4. It provides the complexity C d of the basis (by counting all but the simple points in the associated minimal Besicovitch arrangement) and it also enables to determine the points multiplicities distribution in F _d

of this arrangement. It is available at https://pypi.python.org/pypi/Comp_lib/1.1. Table 1 in Appendix gathers the first 370 values of C d .

2 Complexity versus number of all but simple points in randomly selected arrangements

Let us denote by A d the expected value of the number of all but the simple

points in a randomly chosen minimal Besicovitch arrangement in F _d

. Thanks

to the proof of Theorem 1. in [2], we have:

A d = d

− d(d + 1)(1 − 1 d ) ^d

= (1 − 1

e )d

− 1

2e d + O(1), as d → ∞.

Figure 1 shows the values of ^C

_d

for the first 370 prime numbers.

500 1000 1500 2000 2500

− 5

− 4

− 3

− 2

− 1 1 2 3

0

d C

−A

d

Linear regression line:y= 4.94×10⁻⁵x−0.913

Figure 1: The 370 values of the function that relates each prime number d to

C

d .

2.1 A first test

From the 370 values of Figure 1, we plot the regression line: its slope s is approximately 4.94 × 10

and its intercept is approximately −0.913.

Let us consider the following null hypothesis H

: s = 0. We have to calculate T = ^s−

σ

, where ˆ σ s is the estimated standard deviation of the slope. We obtain ˆ

σ s ≈ 8.74 × 10

and T ≈ 0.565. T follows a student’s t-distribution with (370 − 2) degrees of freedom (see [3, Proposition 1.8]). The acceptance region of the hypothesis test with a 5% risk is approximately [−1.967, 1.967]. Thus it can be concluded that we cannot reject the null hypothesis : the fact that the slope is not significantly different from zero can not be rejected.

2.2 A second test

Figure 2 below shows the distribution of the values of ^C

_d

for the first 370

prime numbers. In regard to the resulting histogram, one may wonder whether

these values are normally distributed or not.

− 5 − 4 − 3 − 2 − 1 1 2 3 5

10 15 20 25 30 35

0

C

−A

d F requency

Figure 2: Distribution of the values of ^C

_d

.

From the result of the first test, we would consider in this part that the function that maps d onto ^C

_d

behaves like a random variable with an ex- pected value Λ close to −0.856. On that assumption we will verify whether the values of ^C

_d

are normally distributed for d ∈ [2, 2531] ∩ N (the null hypoth- esis) or not. For this purpose we use the Shapiro–Wilk test (see [8]). The test statistic W is about 0.991. The associated p-value being about 0.0296, it can be concluded that we can reject the null hypothesis, i.e. the values of ^C

_d

are significantly not normally distributed for d ∈ [2, 2531] ∩ N .

2.3 A third set of tests

Once more, from the result of the first test, we would consider in this part that the function that maps d onto ^C

_d

behaves like a random variable with an expected value Λ close to −0.856 and with a symmetric probability distribution.

On that assumption we will verify whether the values higher than Λ and those smaller than Λ are randomly scattered over the ordered absolute values of ^C

_d

(the null hypothesis) or not. To this end we use a non-parametric test, the Mann–Whitney U test: we determine the ranks of | ^C

_d

| for each d in the considered interval (see [9] or [5]). The ranks sum of the values higher than Λ is approximately normally distributed. The value of U

is about −0.911.

The acceptance region of the hypothesis test with a 5% risk being approximately [−1.960, 1.960], it can be concluded that we cannot reject the null hypothesis, i.e.

the fact that the greater and smaller than Λ values of ^C

_d

for d ∈ [2, 2531] ∩ N are randomly scattered: the symmetry of the probability distribution of our potential pseudorandom variable can not be rejected.

Once more, on our first assumption, we will verify whether the values higher

than Λ and those smaller than Λ are randomly scattered over the first 370 prime

numbers (the null hypothesis) or not. To this end we use the same test, the

Mann–Whitney U test. The prime number ranks sum of the values higher than

Λ is approximately normally distributed. The value of U

is about −0.397. It

can be concluded that we cannot reject the null hypothesis, i.e. the fact that the

greater and smaller than Λ values of ^C

_d

for d ∈ [2, 2531] ∩ N are randomly scattered over the first 370 prime numbers.

2.4 Perspective

Both first test and set of tests could not invalidate the fact that the function that maps d onto ^C

_d

seem to behave like a random variable with Λ as expected value. If we succeed in proving such a statement, we could consider the following unbiased estimator of C d , denoted by C c d :

C c d = A d + Λd

= d

− d(d + 1)(1 − 1

d ) ^d + Λd

= (1 − 1

e )d

+ (Λ − 1

2e )d + o(d), as d → ∞, thanks to the proof of [2, Theorem 1.].

3 Complexity versus number of all but simple points in particular arrangements

3.1 Further details on the minimal Besicovitch arrange- ment providing the complexity

In this part, we will consider particular minimal Besicovitch arrangements which share several geometrical properties with the arrangements considered to deter- mine the complexity and we will compare the expected values of the number of all but simple points in such randomly selected arrangements with C d values.

Before reviewing the whole cycles highlighted in section 1, let us make a quick remark:

Remark 3.1. If a line in an orbit passes through (0, 0) ∈ F _d

all the other lines of this orbit also pass through this point, the elements of the group Γ acting on the lines being in GL

( F _d ).

In section 1 two cases appear, for d ≥ 5: the cases where d ≡ 1 mod 3 and those where d ≡ 2 mod 3.

In both cases, the intercepts of the lines in {L

, L

, L

} are 0 (we have p(0) = 0 thanks to equality 1, Remark 3.1 allowing us to conclude).

The intercepts of the lines in {L

, L

2

, L

} are non zero values, except for d = 1093, the first Wieferich prime number, for which lines intercepts are all zero: p(2) = 0 ⇐⇒

_d

(see equality 1, Remark 3.1 and [4]).

If d ≡ 1 mod 3, the intercepts of the lines in {L ω , L ω

} are 0:

p(ω) = (ω + 1) ^d − ω ^d − 1

d = −(ω ^d )

− ω ^d − 1 d

= − −(ω)

− ω − 1

d = 0,

using the fact that ω is a primitive cubic root of unity in F _d and using Fermat’s

little theorem.

In this part, we will only consider the values of d ∈ [2, 2531] ∩ N where all lines in the 6-cycles do not pass through (0, 0); for the 152 values of d verifying this constraint and also d ≡ 1 mod 3, we denote by M _d

the expected value of the number of all but the simple points in a randomly chosen arrangement sharing geometrical properties with the arrangement providing the complexity;

for the 153 values of d verifying the same constraint and also d ≡ 2 mod 3, we denote by M _d

the similar expected value. Table 1 shows the values of d being in either the first or the second case.

3.2 Lines intersections of the different cycles

The five functions in Γ, other than the identity function Id, will be denote as in [7]:

∀(x, y) ∈ F _d

,

ι(x, y) = (y, x) θ(x, y) = (y − x, −x) θ

(x, y) = (−y, x − y) κ(x, y) = θ ◦ ι(x, y) = (x − y, −y) λ(x, y) = ι ◦ θ(x, y) = (−x, y − x).

Note that ι, κ and λ are of order 2, and θ and θ

are of order 3. We can also easily verify that the fixed points of ι are those of the line L

, the fixed points of κ are those of the line L

and the fixed points of λ are those of the line L

. The following proposition can thus be enunciated:

Proposition 3.2. ∀γ ∈ {ι, κ, λ} and ∀a ∈ F _d \ {0, −1}, if L a and γ(L a ) are two distinct lines, then their intersection point is in line of the fixed points of γ.

Proof. The image of a point under a fonction in Γ ⊂ GL

( F _d ) is a point. So,

∀γ ∈ {ι, κ, λ} and ∀a ∈ F _d \ {0, −1}, if L a and γ(L a ) are two distinct lines, i.e.

if their intersection is a point:

γ(L a ∩ γ(L a )) = γ(L a ) ∩ γ(γ(L a ))

= L a ∩ γ(L a ),

each of the considered functions being of order 2. The point L a ∩ γ(L a ) is thus in the fixed line of γ.

Let us henceforth denote by T the set F _d

\ {L

, L

, L

}. In each 6-cycle, for all γ ∈ Γ and for all a ∈ F _d (such that L a is in the considered 6-cycle), L a and γ(L a ) are distinct; we can therefore apply Proposition 3.2: in the case where all the lines in a 6-cycle do not pass through (0, 0) (the prevalent selected case in subsection 3.1), there exist 3 intersection points of the 6-cycle lines on each line of {L

, L

, L

}:

• on L

: L a ∩ κ(L a ), θ(L a ) ∩ λ(L a ) and θ

(L a ) ∩ ι(L a );

• on L

: L a ∩ λ(L a ), θ(L a ) ∩ ι(L a ) and θ

(L a ) ∩ κ(L a );

• on L

: L a ∩ ι(L a ), θ(L a ) ∩ κ(L a ) and θ

(L a ) ∩ λ(L a ).

An other proposition can be added:

Proposition 3.3. In the case where all the lines in a 6-cycle do not pass

through (0, 0), two of the described above 6-cycle intersection points on a line of

{L

, L

, L

} do not coincide.

Proof. Let us consider a 6-cycle. Its lines do not pass through the origin. a ∈ F _d , such that L a is in this 6-cycle. We assume that L a ∩ κ(L a ) and θ(L a ) ∩ λ(L a ) coincide on L

. Knowing that θ(L

) = L

(see [7]), we have:

θ(L a ∩ κ(L a ) ∩ θ(L a ) ∩ λ(L a )) ∈ L

θ(L a ) ∩ λ(L a ) ∩ θ

(L a ) ∩ ι(L a ) ∈ L

.

So θ(L a )∩λ ∈ L

∩L

= (0, 0); it contradicts the hypothesis of the proposition.

The considered points do not coincide.

All the other cases can be demonstrated in the same way.

Thus the remaining 6 intersection points of the 6-cycle lines are in T . We can finally prove the following proposition (in the case where d ≥ 11, otherwise there is no 6-cycle in the arrangement L ):

Proposition 3.4. The 6 remaining point in T (in the case where all the lines in the 6-cycle do not pass through (0, 0)) are distinct.

Proof. We first prove the following lemma:

Lemma 3.5. θ has a single fixed point in F _d

⇐⇒ d 6= 3.

Proof. θ ∈ GL

( F _d ) then (0, 0) is a fixed point of θ.

For (x, y) ∈ F _d

:

θ(x, y) = (x, y) ⇐⇒ y − x = x and − x = y

⇐⇒ 3x = 0 and y = −x.

The result follows.

Let us consider a 6-cycle. Its lines do not pass through the origin. a ∈ F _d , such that L a is in this 6-cycle.

Let us assume that 3 lines in the 6-cycle are concurrent in P ∈ T . It is clear from the foregoing that these lines are whether L a , θ(L a ) and θ

(L a ) or ι(L a ), κ(L a ) and λ(L a ). We have:

θ(P) = θ(L a ∩ θ(L a ) ∩ θ

(L a )) or θ(ι(L a ) ∩ κ(L a ) ∩ λ(L a ))

= θ(L a ) ∩ θ

(L a ) ∩ L a or κ(L a ) ∩ λ(L a ) ∩ ι(L a ) .

In both cases θ(P) = P i.e. P is a fixed point of θ. It means that P = (0, 0) thanks to Lemma 3.5, knowing that d ≥ 11; it contradicts the hypothesis of the proposition. The 6 remaining point in T are distinct.

The cases of {L

, L

2

, L

} and {L ω , L _ω

} can be considered as degenerate cases of a 6-cycle. Let us focus on the first arrangement. From [7], we get ι(L

) = L

, λ(L

) = L

2

and κ(L

) = L

2

. Thanks to Proposition 3.2, the 3 intersection points of lines in {L

, L

2

, L

} are:

• on L

: L

∩ L

2

;

• on L

: L

∩ L

2

;

• on L

: L

∩ L

.

We note that this result is just a particular case of the above result.

The Figure 3 below provides two examples of minimal Besicovitch arrange- ments leading to the determination of the complexity. For the first one (d = 7), we are in the case where d ≡ 1 mod 3, for the second one (d = 11) in the case where d ≡ 2 mod 3. The above results and in particular those of Propositions 3.2, 3.3 and 3.4 are emphasised.

F

F

Figure 3: Lines of the minimal Besicovitch arrangement in F _d

providing the complexity C d where d = 7 (on the left) and d = 11 (on the right). Red lines are those of {L

, L

, L

}, green ones are those of {L

, L

2

, L

}, blue ones are those of {L ω , L _ω

} and black ones are lines of a 6-cycle. The number of all but the simple points is 25 for d = 7, and 67 for d = 11; thus C

= 25 and C

= 67.

3.3 The first model

We first consider the case where d ≡ 1 mod 3. Let us denote by Ω

the set of minimal Besicovitch arrangements verifying some geometrical constraints simi- lar to those of the considered Besicovitch arrangements. In such arrangements:

• there exist 3 lines of equations x = 0, y = 0 and y = x (let us denote by l a this lines set);

• there exist 2 lines that pass through the origin (let us denote by l

this lines set);

• there exist 3 lines that do not pass through the origin, their 3 intersection points being respectively on each of the 3 lines in l a (let us denote by l

this lines set);

• there exist ^d−

sets of 6 lines, all verifying the same constraints as in Propositions 3.2, 3.3 and 3.4.

In order to calculate the average number of all but simple points in such

arrangements, we will build a probability space: Ω

. The σ-algebra chosen here

is the finite collection of all subsets of Ω

. Our probability measure, denoted by P, assigns equal probabilities to all outcomes.

For Q in F _d

, let M Q be the random variable that maps A ∈ Ω

to the multiplicity of Q in A.

With the aim of knowing the expected number of simple points in such particular arrangements, we determine P(M Q = 1), for all Q in F _d

. Two cases appear: either Q is in a line of l a (apart from the origin) or not.

3.3.1 Q is in a line of l a (apart from the origin) In this case, for A ∈ Ω

, we have:

M Q (A) = 1 ⇐⇒ none of the d − 2 lines of A (other than those of l a ) pass through Q.

We already know that lines of l

do not pass through this point.

There is a ^d−2 _d−1 × ^d−3 _d−2 probability that the two distinct intersection points between lines of l

and the considered line of l a do not coincide with Q (a line is composed of d points and the origin is here not considered).

Similarly, there is a ^d− _d−

× ^d− _d−

× ^d− _d−

probability that the three distinct intersection points between lines of a set of 6 lines (verifying the same constraints as in Propositions 3.2 and 3.3) and the considered line of l a do not coincide with Q.

Finally, considering the ^d−

sets of 6 lines and the lines in l

and l

, we obtain in this case:

P(M Q = 1) = d − 3

d − 1 × d − 4 d − 1

.

3.3.2 Q is not in a line of l a

In this case, for A ∈ Ω

, we have:

M Q (A) = 1 ⇐⇒ exactly one line of the d − 2 lines of A (other than those of l a ) passes through Q.

We will use the following results to study in more detail the different sub- cases. In F _d

\ l a , there are d

− 3 × (d − 1) − 1 = d

− 3d + 2 points. In F _d

\ l a ∪ l

, there are 2(d − 1) points of multiplicity 1 and the remaining points of multiplicity 0 (d

−5d + 4 points). In F _d

\ l a ∪ l

, there are 3(d − 3) points of multiplicity 1 and the remaining points of multiplicity 0 (d

− 6d + 11 points).

In the union of F _d

\ l a and a 6 lines set, there are 6(d− 5) points of multiplicity 1, 6 points of multiplicity 2 (see Proposition 3.4) and the remaining points of multiplicity 0 (d

− 9d + 26 points).

This case can be divided in 3 subcases:

• the first one where the line that passes through Q is in l

; then the prob- ability is:

2(d − 1)

d

− 3d + 2 × d

− 6d + 11

d

− 3d + 2 × d

− 9d + 26 d

− 3d + 2

;

• the second one where the line that passes through Q is in l

; then the probability is:

d

− 5d + 4

d

− 3d + 2 × 3(d − 3)

d

− 3d + 2 × d

− 9d + 26 d

− 3d + 2

;

• the third one where the line that passes through Q is in one of the ^d−

sets of 6 lines; then the probability is:

d

− 5d + 4

d

− 3d + 2 × d

− 6d + 11

d

− 3d + 2 × d − 7 6

6(d − 5) d

− 3d + 2

d

− 9d + 26 d

− 3d + 2

.

Hence we have in this specific case:

P(M Q = 1) = 2

d − 2 × d

− 6d + 11

d

− 3d + 2 × d

− 9d + 26 d

− 3d + 2

+ d − 4

d − 2 × 3(d − 3)

d

− 3d + 2 × d

− 9d + 26 d

− 3d + 2

+ d − 4

d − 2 × d

− 6d + 11

d

− 3d + 2 × d

− 12d + 35 d

− 3d + 2

d

− 9d + 26 d

− 3d + 2

.

3.3.3 The expected value of M _d

Recall that our aim is to determine the expected value M _d

of the number of all but simple points in arrangements of Ω

in order to compare it with the value of the complexity C d .

Thanks to the results of the above section and knowing that the first case concerns 3d − 3 points and the second one d

− 3d + 2 points, we get:

M _d

=d

−

3(d − 3) × d − 4 d − 1

+ 2(d

− 6d + 11)

d − 2 × d

− 9d + 26 d

− 3d + 2

+ 3(d − 3)(d − 4)

d − 2 × d

− 9d + 26 d

− 3d + 2

+ (d − 4)(d

− 6d + 11)

d − 2 × d

− 12d + 35 d

− 3d + 2

d

− 9d + 26 d

− 3d + 2

.

Using the Computer Algebra System Giac/Xcas (Parisse and De Graeve, 2017, http://www-fourier.ujf-grenoble.fr/~parisse/giac_fr.html , ver- sion 1.2.3), we obtain:

M _d

= (1 − 1

e )d

+ 1

e − 3 exp(− 1 2 )

d + O(1), as d → ∞.

3.4 The second model

We henceforth consider the case where d ≡ 2 mod 3. Let us denote by Ω

the set of minimal Besicovitch arrangements verifying some geometrical con- straints similar to those of the considered Besicovitch arrangements. In such arrangements:

• there exist 3 lines of equations x = 0, y = 0 and y = x (let us denote by

l a this lines set);

• there exist 3 lines that do not pass through the origin, their 3 intersection points being respectively on each of the 3 lines in l a (let us denote by l

this lines set);

• there exist ^d−5

sets of 6 lines, all verifying the same constraints as in Propositions 3.2, 3.3 and 3.4.

In order to calculate the average number of all but simple points in such arrangements, we will build a probability space: Ω

. The σ-algebra chosen here is the finite collection of all subsets of Ω

. Our probability measure, denoted by P , assigns equal probabilities to all outcomes.

For Q in F _d

, let M Q be the random variable that maps A ∈ Ω

to the multiplicity of Q in A.

With the aim of knowing the expected number of simple points in such particular arrangements, we determine P (M Q = 1), for all Q in F _d

. Two cases appear: either Q is in a line of l a (apart from the origin) or not.

3.4.1 Q is in a line of l a (apart from the origin) In this case, for A ∈ Ω

, we have:

M Q (A) = 1 ⇐⇒ none of the d − 2 lines of A (other than those of l a ) pass through Q.

There is a ^d− _d−

× ^d− _d−

probability that the two distinct intersection points between lines of l

and the considered line of l a do not coincide with Q.

Similarly, there is a ^d− _d−

× ^d− _d−

× ^d− _d−

probability that the three distinct intersection points between lines of a set of 6 lines and the considered line of l a

do not coincide with Q.

Finally, considering the ^d−

sets of 6 lines and the lines in l

, we obtain in this case:

P(M Q = 1) = d − 3

d − 1 × d − 4 d − 1

.

3.4.2 Q is not in a line of l a

In this case, for A ∈ Ω

, we have:

M Q (A) = 1 ⇐⇒ exactly one line of the d − 2 lines of A (other than those of l a ) passes through Q.

We will use the following results to study in more detail the different sub- cases. In F _d

\ l a , there are d

− 3d + 2 points. In F _d

\ l a ∪ l

, there are 3(d − 3) points of multiplicity 1 and the remaining points of multiplicity 0 (d

− 6d + 11 points). In the union of F _d

\ l a and a 6 lines set, there are 6(d − 5) points of multiplicity 1, 6 points of multiplicity 2 (see Proposition 3.4) and the remaining points of multiplicity 0 (d

− 9d + 26 points).

This case can be divided in 2 subcases:

• the first one where the line that passes through Q is in l

; then the prob- ability is:

3(d − 3)

d

− 3d + 2 × d

− 9d + 26 d

− 3d + 2

;

• the second one where the line that passes through Q is in one of the ^d−

sets of 6 lines; then the probability is:

d

− 6d + 11

d

− 3d + 2 × d − 5 6

6(d − 5) d

− 3d + 2

d

− 9d + 26 d

− 3d + 2

.

Hence we have in this specific case:

P (M Q = 1) = 3(d − 3)

d

− 3d + 2 × d

− 9d + 26 d

− 3d + 2

+ d

− 6d + 11

d

− 3d + 2 × d

− 10d + 25 d

− 3d + 2

d

− 9d + 26 d

− 3d + 2

.

3.4.3 The expected value of M _d

Recall that our aim is to determine the expected value M _d

of the number of all but simple points in arrangements of Ω

in order to compare it with the value of the complexity C d .

Thanks to the results of the above section and knowing that the first case concerns 3d − 3 points and the second one d

− 3d + 2 points, we get:

M _d

=d

−

3(d − 3) × d − 4 d − 1

+ 3(d − 3) × d

− 9d + 26 d

− 3d + 2

+ (d

− 6d + 11) × d

− 10d + 25 d

− 3d + 2

d

− 9d + 26 d

− 3d + 2

.

Using the Computer Algebra System Xcas, we obtain:

M _d

= (1 − 1

e )d

+ 1

e − 3 exp(− 1 2 )

d + O(1), as d → ∞.

3.5 Results

Figure 4 shows values of both ^C

_d

and ^C

_d

for the selected prime num- bers d.

500 1000 1500 2000 2500

−4

−3

−2

−1 1 2 3 4 5

0

C

d

d

Linear regression line:y= 1.94×10⁻⁴x+ 0.352

500 1000 1500 2000 2500

−4

−3

−2

−1 1 2 3 4 5

0

Linear regression line:y=−1.88×10⁻⁴x+ 0.511

d C

d

Figure 4: Values of both ^C

_d

(on the left) and ^C

_d

(on the right) for the

selected prime numbers d.

3.5.1 A first test in each case

From the 152 left plotted values on Figure 4, we draw the regression line: its slope s

is approximately 1.94 × 10

and its intercept is approximately 0.352.

Let us consider the following null hypothesis H

: s

= 0. We have to calculate T

= ^s

σ

, where ˆ σ s

is the estimated standard deviation of the slope. We obtain ˆ σ s

≈ 1.35 × 10

and T

≈ 1.43. T

follows a student’s t-distribution with (152 − 2) degrees of freedom [3, Proposition 1.8]. The acceptance region of the hypothesis test with a 5% risk is approximately [−1.976, 1.976]. Thus it can be concluded that we cannot reject the null hypothesis, i.e. the fact that the slope s

is not significantly different from zero.

From the 153 right plotted values on Figure 4, we draw the regression line:

its slope s

is approximately −1.88 × 10

and its intercept is approximately 0.511. Let us consider the following null hypothesis H

: s

= 0. We again have to calculate T

= ^s

_σ

s∗∗

. We here obtain ˆ σ s

≈ 1.25 × 10

and T

≈

−1.50. T

follows a student’s t-distribution with (153 − 2) degrees of freedom.

The acceptance region of the hypothesis test with a 5% risk is approximately [−1.976, 1.976]. Thus it can be concluded that we cannot reject the fact that the slope s

is not significantly different from zero.

3.5.2 A set of tests in each case

Figure 5 below shows the distribution of the values of ^C

_d

(on the left) and

C

d (on the right) for the considered values of d.

− 3 − 2 − 1 1 2 3 4 2

4 6 8 10 12 14 16 18

0

C

d

F requency

− 4 − 3 − 2 − 1 1 2 3 2

4 6 8 10 12 14 16 18

0

C

d

F requency

Figure 5: Distribution of values of both ^C

_d

(on the left) and ^C

_d

(on the right) for the selected prime numbers d.

From the result of the first test in section 3.5.1, we would consider in this

part that the function that maps d onto ^C

_d

behaves like a random variable

with an expected value Λ

close to 0.576 and with a symmetric probability

distribution (for the considered values of d). On that assumption we will verify

whether the values higher than Λ

and those smaller than Λ

are randomly

scattered over the ordered absolute values of ^C

_d

(the null hypothesis) or

not. To this end we use a non-parametric test, the Mann–Whitney U test: we determine the ranks of | ^C

_d

| for each d in the considered interval (see [9]

or [5]). The ranks sum of the values higher than Λ

is approximately normally distributed. The value of U

is about −0.673. The acceptance region of the hypothesis test with a 5% risk being approximately [−1.960, 1.960], it can be concluded that we cannot reject the null hypothesis, i.e. the fact that the greater and smaller than Λ

values of ^C

_d

are randomly scattered: the symmetry of the probability distribution of this potential pseudorandom variable can not be rejected.

On the same assumption, we will also verify whether the values higher than Λ

and those smaller than Λ

are randomly scattered over the consid- ered prime numbers (the null hypothesis) or not. To this end we again use the Mann–Whitney U test. The prime numbers ranks sum of the values higher than Λ

is approximately normally distributed. The value of U

is about 0.721. It can be concluded that we cannot reject the null hypothesis, i.e. the fact that the greater and smaller than Λ

values of ^C

_d

are randomly scattered over the considered prime numbers.

From the result of the second test in 3.5.1, we would consider in this part that the function that maps d onto ^C

_d

behaves like a random variable with an expected value Λ

close to 0.297 and with a symmetric probability distribution (for the considered values of d). On that assumption we will verify whether the values higher than Λ

and those smaller than Λ

are randomly scattered over the ordered absolute values of ^C

_d

(the null hypothesis) or not. To this end we again use the Mann–Whitney U test. The value of U

is here about −1.08.

It can once more be concluded that we cannot reject the fact that the greater and smaller than Λ

values of ^C

_d

are randomly scattered: the symmetry of the probability distribution of this potential pseudorandom variable can not be rejected.

On the same assumption, we will verify whether the values higher than Λ

and those smaller than Λ

are randomly scattered over the considered prime numbers or not. To this end we again use the Mann–Whitney U test. The value of U

is here about −1.77. It can once more be concluded that we cannot reject the fact that the greater and smaller than Λ

values of ^C

_d

are randomly scattered over the considered prime numbers.

3.5.3 Perspective

Both first test and set of tests could not invalidate the fact that the function

that maps d onto ^C

_d

and the one that maps d onto ^C

_d

seem to behave

like random variables with respectively Λ

and Λ

as expected values. Λ

and

Λ

are both positive numbers, whereas Λ is negative; the added geometrical

constraints seem to reduce in average the number of all but the simple points

generated by a randomly chosen minimal Besicovitch arrangement. This reduc-

tion is slightly highter than expected. Our arrangements cannot obviously be

limited to the considered geometrically constrained arrangement. Adding con-

straints for better modeling the arrangements and finding a way to determine

whether the considered functions could be considered as high-quality pseudo-

random number generators (PRNG) sketch some avenues for future research on

the subject.

Appendix

d 2 3 5∗∗ 7∗ 11∗∗ 13∗ 17∗∗ 19∗ 23∗∗ 29∗∗ 31∗ 37∗ 41∗∗ 43∗ 47∗∗ 53∗∗ 59 61∗ 67∗ 71∗∗

Cd 1 6 13 25 67 100 163 229 334 448 625 844 1075 1114 1402 1786 1912 2218 2752 3046 d 73∗ 79 83 89∗∗ 97∗ 101∗∗ 103∗ 107∗∗ 109∗ 113∗∗ 127∗ 131∗∗ 137∗∗ 139∗ 149∗∗ 151∗

Cd 3307 3685 4189 4972 5971 6367 6475 7102 7315 8107 10150 10879 11824 12220 13936 14176 d 157∗ 163∗ 167∗∗ 173∗∗ 179 181∗ 191∗∗ 193 197∗∗ 199∗ 211∗ 223∗ 227 229∗ 233∗∗

Cd 15529 16546 17440 18799 19789 20758 22945 23251 24430 24739 28186 31348 32482 33127 33721 d 239∗∗ 241∗ 251∗∗ 257∗∗ 263∗∗ 269∗∗ 271∗ 277∗ 281∗∗ 283∗ 293∗∗ 307∗ 311∗∗ 313∗ 317∗∗

Cd 35800 36577 39808 41515 43795 45214 45940 48160 49507 49747 54625 59248 60886 60592 63535 d 331∗ 337 347∗∗ 349∗ 353∗∗ 359∗∗ 367∗ 373∗ 379∗ 383∗∗ 389∗∗ 397∗ 401∗∗ 409∗ 419 Cd 68794 71359 74710 76915 78466 81265 84772 87586 90232 92203 95716 99352 101314 104797 109873

d 421 431∗∗ 433∗ 439∗ 443 449∗∗ 457 461∗∗ 463∗ 467∗∗ 479∗∗ 487∗ 491∗∗

Cd 111913 117079 118249 122023 123148 127207 130669 133840 134125 136486 144355 150223 151696 d 499∗ 503∗∗ 509∗∗ 521∗∗ 523∗ 541∗ 547 557∗∗ 563∗∗ 569∗∗ 571∗ 577∗ 587∗∗

Cd 157138 159607 162508 171607 172345 183730 188854 195535 200263 204214 203680 210088 216331 d 593∗∗ 599∗∗ 601 607∗ 613∗ 617∗∗ 619 631∗ 641∗∗ 643∗ 647∗∗ 653∗∗ 659∗∗

Cd 221269 226318 227140 232981 237046 239626 242398 250861 259405 260467 263722 268363 273217 d 661∗ 673∗ 677∗∗ 683∗∗ 691 701 709∗ 719∗∗ 727∗ 733∗ 739∗ 743∗∗ 751∗

Cd 275827 286606 288166 294208 299602 312463 319282 325690 332941 338929 344065 347074 353806 d 757 761∗∗ 769∗ 773∗∗ 787 797∗∗ 809∗∗ 811∗ 821∗∗ 823∗ 827∗∗ 829∗ 839∗∗

Cd 360034 364345 373825 377044 390112 400093 413593 416320 424864 425239 431245 436477 443629

d 853∗ 857 859∗ 863∗∗ 877∗ 881∗∗ 883∗ 887 907 911 919∗ 929 937∗

Cd 458275 463174 466087 472573 483487 488704 491626 494824 519175 523180 533941 543892 553420 d 941∗∗ 947∗∗ 953∗∗ 967∗ 971 977 983∗∗ 991∗ 997∗ 1009∗ 1013∗∗ 1019∗∗ 1021∗

Cd 559363 565651 574390 589471 594424 599923 610498 620311 627001 644440 644356 653449 658795 d 1031∗∗ 1033∗ 1039 1049∗∗ 1051∗ 1061∗∗ 1063∗ 1069∗ 1087∗ 1091 1093 1097∗∗ 1103∗∗

Cd 671311 674257 680911 692635 697756 710902 712840 723076 745966 752482 752740 759808 768805 d 1109 1117∗ 1123∗ 1129∗ 1151∗∗ 1153∗ 1163∗∗ 1171∗ 1181∗∗ 1187∗∗ 1193 1201∗ 1213∗

Cd 779941 787798 794254 806077 837823 838891 851632 862882 878656 887017 900982 911497 929935 d 1217 1223 1229∗∗ 1231 1237 1249∗ 1259 1277∗∗ 1279∗ 1283 1289 1291∗

Cd 936253 943267 956872 956560 964465 985237 1000621 1029562 1033756 1039588 1047226 1052251 d 1297∗ 1301∗∗ 1303 1307∗∗ 1319∗∗ 1321∗ 1327∗ 1361∗∗ 1367∗∗ 1373∗∗ 1381∗ 1399∗

Cd 1063438 1068115 1071913 1078375 1101274 1102360 1113577 1169632 1181578 1192081 1205287 1235425 d 1409∗∗ 1423∗ 1427∗∗ 1429∗ 1433∗∗ 1439 1447∗ 1451∗∗ 1453∗ 1459∗ 1471∗ 1481∗∗

Cd 1251661 1280677 1281337 1291003 1294351 1303495 1326448 1329037 1330435 1344493 1364623 1385854 d 1483∗ 1487 1489 1493 1499∗∗ 1511∗∗ 1523∗∗ 1531 1543∗ 1549∗ 1553∗∗ 1559∗∗

Cd 1387588 1398910 1400191 1407457 1420246 1444459 1464190 1477492 1502308 1513120 1524895 1533844 d 1567∗ 1571∗∗ 1579∗ 1583∗∗ 1597∗ 1601∗∗ 1607∗∗ 1609∗ 1613 1619∗∗ 1621∗ 1627∗

Cd 1549756 1558054 1571542 1584523 1609036 1614859 1630015 1638106 1644892 1655251 1659781 1673800 d 1637 1657 1663∗ 1667∗∗ 1669∗ 1693∗ 1697∗∗ 1699∗ 1709∗∗ 1721∗∗ 1723∗ 1733∗∗

Cd 1692076 1735675 1746766 1755874 1759345 1811281 1817827 1821148 1845148 1868239 1875610 1893445

d 1741∗ 1747∗ 1753∗ 1759∗ 1777∗ 1783∗ 1787∗∗ 1789∗ 1801∗ 1811 1823∗∗ 1831∗

Cd 1915426 1926787 1938808 1956460 1991359 2006128 2018023 2019100 2055712 2074435 2092648 2118334 d 1847 1861∗ 1867∗ 1871∗∗ 1873∗ 1877∗∗ 1879∗ 1889∗∗ 1901 1907∗∗ 1913∗∗ 1931∗∗

Cd 2156626 2185264 2198473 2211484 2216392 2224747 2228053 2253946 2281783 2297935 2303611 2355019 d 1933∗ 1949∗∗ 1951∗ 1973∗∗ 1979∗∗ 1987∗ 1993 1997 1999∗ 2003 2011∗ 2017∗

Cd 2356819 2398531 2407693 2459041 2474182 2493151 2513734 2520214 2525929 2534818 2554063 2571514 d 2027∗∗ 2029∗ 2039∗∗ 2053∗ 2063∗∗ 2069∗∗ 2081∗∗ 2083∗ 2087 2089 2099∗∗ 2111∗∗

Cd 2594968 2605618 2625871 2661322 2685235 2700313 2739367 2741827 2750443 2757349 2783251 2809579 d 2113 2129∗∗ 2131∗ 2137∗ 2141∗∗ 2143∗ 2153∗∗ 2161∗ 2179∗ 2203∗ 2207∗∗ 2213∗∗

Cd 2821639 2862883 2869180 2886973 2892547 2898181 2928235 2952049 3001276 3069025 3075811 3092218 d 2221∗ 2237∗∗ 2239∗ 2243 2251 2267∗∗ 2269∗ 2273∗∗ 2281∗ 2287∗ 2293∗ 2297∗∗

Cd 3114424 3159310 3166807 3175720 3199828 3244723 3256783 3265912 3285589 3303373 3326029 3330658 d 2309∗∗ 2311 2333∗∗ 2339∗∗ 2341∗ 2347∗ 2351∗∗ 2357∗∗ 2371∗ 2377 2381∗∗ 2383∗

Cd 3372679 3373075 3434839 3457402 3462010 3480025 3497599 3510505 3555751 3567400 3579163 3587740 d 2389 2393∗∗ 2399∗∗ 2411∗∗ 2417∗∗ 2423 2437 2441∗∗ 2447∗∗ 2459∗∗ 2467∗ 2473∗

Cd 3602248 3614269 3636025 3671155 3687757 3709300 3749929 3761812 3780007 3821119 3847576 3861457 d 2477 2503∗ 2521∗ 2531∗∗

Cd 3878440 3960268 4014841 4046863

Table 1: The complexities values. Values of d with one asterisk correspond to arrangements where d ≡ 1 mod 3 and where all the lines (except those of {L

, L

, L

} and {L ω , L ω

}) do not pass through the origin, whereas values of d with two asterisks correspond to arrangements where d ≡ 2 mod 3 and where all the lines (except those of {L

, L

, L

}) do not pass through the origin.

References

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_q . Journal of Number Theory, 180C:533–543, 2017.

[3] P.-A. Cornillon and E. Matzner-Lober. R´egression : Th´eorie et applications.

Springer, 2007. 302 p.

[4] F. G. Dorais and D. Klyve. A Wieferich prime search up to 6.7×10

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[6] A. J. Menezes, I. F. Blake, S. Gao, R. C. Mullin, S. A. Vanstone, and T. Yaghoobian, editors. Applications of Finite Fields. Kluwer Academic Publishers, 1993.

[7] E.J. Pickett and S. Vinatier. About the complexity of cyclotomic self dual normal bases. In preparation.

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[9] F. Wilcoxon. Individual comparisons by ranking methods. Biometrics Bul-

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